Activity 10: Cumulative frequency
Framework reference: Page 259
Strand:
Topic:
Handling data.
Processing and representing
data.
Pupils should be taught to: Calculate
statistics from the data, using ICT
as appropriate.
Year group: 9
Objectives: Estimate the median and
interquartile range for a large set
of grouped data.
Key Vocabulary: Median, interquartile range,
cumulative frequency diagram,
boxplot.
Resources required: Class set of calculators
plus Viewscreenor TI-SmartView
emulator
Summary
This is very much a teacher-directed session
and uses quite advanced calculator skills.
It assumes that the students have already come
across grouping data into class intervals.
The lesson goes through the following stages:
Generate data → Frequency table →
Cumulative frequency table → Cumulative
frequency graph. Students then use the
cumulative frequency graph to read off the
values of the quartiles and the median.
Optionally, they will see another method of
finding quartiles and median – using a ‘boxplot’.
Instructions for the teacher
(1) Revise grouping data into class intervals, by
working through Activities 1 and 2 using the
demo calculator. Explain how to calculate 1-Var
summaries of data from a single list (list L1) and
frequency data from two lists (lists L2 and L3).
The method described in Activity 1 of the
handout involves storing 100 randomly chosen
integer scores, modelled by a normal distribution
Of course, there is no need for students to
understand how or why this works--it is simply a
way of getting the same set of fairly realistic
data into all the calculators. (An alternative
method would be to use link cables.)
(2) Distribute the calculators and handouts and
ask students to work through Activities 1 – 3.
The completed table in Activity 2 is shown in the
mext column.
Teachers Teaching with TechnologyTM
Teacher’s notes
Interval
0-9
10 - 19
20 - 20
30 - 39
40 - 49
50 - 59
60 - 69
70 - 79
80 - 89
90 - 99
Mid-points
4.5
14.5
24.5
34.5
44.5
54.5
64.5
74.5
84.5
94.5
Frequency
1
2
7
12
19
26
23
7
3
0
(3) Discuss with the class their responses to
Activity 3(c). The summaries are as follows
L1
L2,L3
Min
6
4.5
Q1
41.5
44.5
Med
54
54.5
Q3
61
64.5
Max
87
84.5
The answers are different because transforming
the original data into a frequency table involved
a loss of information. Summaries of frequency
data are based on the simplifying assumption
that all the values are centred on the interval
that contains them. So, scores like 31, 61, 72,
41, etc. will not match the frequency data well,
whereas marks like 45, 74, 36, 65 etc. will.
(4) Using the demo calculator, lead the class
carefully through Activities 4 and 5, emphasising
that this is a ‘less than’ cumulative frequency
graph. i.e. it indicates how many students
scored less than the value plotted on the X
axis. You will probably need to explain quartiles
and how they can be read from a cumulative
frequency graph. (Note that in Activity 5 taking
horizontal values of 25, 50 and 75 will not be
exactly correct – for example the median is
halfway between the 50th and 51st value – but
with a batch as large as this the error is
negligible and can be ignored.)
(5) Then ask the class to tackle Activities 4 – 5.
(6) Boxplots provide a simple and intuitive
means of dealing with the median and quartiles,
Handout 3 provides an extension activity in
which a boxplot of the data in L1 is superimposed on the cumulative frequency graph.
This activity was first published in
30 Calculator Lessons for Key Stage 3
(A+B Books).
Calculator Maths: Handling Data, p 40
provides activities to introduce students to
boxplots.
Activity 10: Cumulative frequency
1)
Handout 1
Data
You can make your calculator create 100 numbers that could
represent the marks of 100 students.
First, reset the calculator’s random facility. Press:
2 ¿ I | 1Í
Now press:
I ~ 5 I | 6 50 ¢ 15 ¢ 100 ¤
¿ydÍ
2)
Frequency table
Now you are going to arrange all the L1 data into intervals of 0-9, 10-19, 20-29 etc.
(a) Draw up a table like this:
Interval
0–9
10–19
20–29
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
Mid-points
4.5
14.5
...
...
...
...
...
...
...
...
...
Frequency
...
...
...
...
...
...
...
...
...
...
...
(b) In the second column enter all the
mid-points of the intervals. (Notice 4.5
is halfway between 0 and 9.)
(c) Sort list L1 into ascending order. To
do this press …
2
y
d
¤
Í
Return to the list screen (press …
1)
and scroll down the re-ordered list L1,
counting the number of scores in each of
the intervals 0-9, 10-19, 20-29, …
Record these frequencies in the third
column of the table.
Use the frequency table to answer the following questions.
(d) How many students scored between 30 and 39 inclusive?
(e) How many students scored between 50 and 69 inclusive?
(f) How many students scored less than 50?
(g) How many students scored less than 70?
(h) On your calculator enter the interval mid-points into
list L2 and the frequencies into list L3.
(3)
Summarising frequency data
(a) Enter the command 1-Var Stats L1 (press …
~ 1
y
d
Í).
Scroll down the screen to find and write down the following summary values for L1.
Minimum,
Lower quartile,
Median,
Upper quartile, Maximum.
(b) Enter the command 1-Var Stats L2,L3 and write down the same five
summary values for the frequency data in L2 and L3.
(c) Write a sentence comparing your answers to parts (a) and (b), explaining any similarities or
differences.
Teachers Teaching with TechnologyTM
Activity 10: Cumulative frequency
(4)
Handout 2
‘Less than’ cumulative frequency table
Now you are going to draw up a table to show the number of students who scored
less than the various marks.
For example, from your frequency table you can see:
1 student
scored less than 10 (i.e in the range 0-9)
3 students (1+2)
scored less than 20 (i.e in the range 0-9, or 10-19), etc.
Enter the values 10, 20, 30, …100 in L4
(these are the ‘less than’ scores).
Return to the home screen and apply the cumSum command to L3,
storing the results in L5, as follows. Press:
y9~6yf¤¿yhÍ
Return to the list screen and move the right cursor so that you can view
lists L4 and L5 together.
(a) How many students scored less than 60?
(b) How many students scored less than 80?
(c) How many students scored between 20 and 49 inclusive?
(5)
Cumulative frequency graph
You can now use the data in L4 and L5 to plot a cumulative frequency graph.
Press y
" 1 to select Plot1 and select
the settings shown.
Choose suitable Window settings and display the
cumulative frequency graph by pressing r.
You can use the cumulative frequency graph to read off the median (Med), lower
quartile (Q1) and upper quartile (Q3). In this example, the batch size is 100, so Q1,
Med and Q3 correspond to scores with horizontal values of roughly 25, 50 and 75.
(a) Return to the home screen and enter Horizontal 25 by pressing:
y < 3 25 Í.
See the effect that this has on the graph. Use the free-floating cursor keys (not r) to
estimate where this horizontal line crosses the cumulative frequency graph.
This is your estimate of the lower quartile: write it down.
(b) With a similar approach, use the lines Horizontal 50 and
Horizontal 75 to estimate the median and the upper quartile.
Write them down.
(c) Compare these estimated values with the answers you
produced in Activity 3.
Teachers Teaching with TechnologyTM
Activity 10: Cumulative frequency
(6)
Extension handout 3
Boxplots
The calculator provides another way of displaying the median and quartiles--using a boxplot.
(a) Use Plot 2 to display a boxplot of the data in L1 on the same
graphing screen. Choose the settings shown here.
Press r and the left and right cursor keys to read off the
five key positions of the boxplot
(minX, Q1, Med, Q3 and maxX).
Write them down.
(b) Work out and write down the value of the interquartile range, Q3 – Q1.
(c) Compare Q1, Med and Q3 values with the values you estimated from
the cumulative frequency graph in Activity 5.
How accurate was the estimate?
(d) The data in L2 and L3 are similar to the data in L1 but they are in a frequency format.
Set up Plot 3 to display a boxplot of the frequency data in L2
and L3 on the same graphing screen. Choose the settings shown
here.
Switch off the cumulative frequency graph (Plot1).
Plot the boxplots of L1 and L2,L3 on the same screens.
Are there differences?
Can you explain them? Think back to what you did in Activity 3.
Teachers Teaching with TechnologyTM